MATHICSE Technical Report : Reduced order methods for uncertainty quantification problems

This work provides a review on reduced order methods in solving uncertainty quantification problems. A quick introduction of the reduced order methods, including proper orthogonal decomposition and greedy reduced basis methods, are presented along with the essential components of general greedy algorithm, a posteriori error estimation and Offline-Online decomposition. More advanced reduced order methods are then developed for solving typical uncertainty quantification problems involving pointwise evaluation and/or statistical integration, such as failure probability evaluation, Bayesian inverse problems and variational data assimilation. Three expository examples are provided to demonstrate the efficiency and accuracy of the reduced order methods, shedding the light on their potential for solving problems dealing with more general outputs, as well as time dependent, vectorial noncoercive parametrized problems.